Journal of Theoretical Probability

, Volume 25, Issue 2, pp 536–564

Asymptotic Theory for Fractional Regression Models via Malliavin Calculus

Article

DOI: 10.1007/s10959-010-0302-y

Cite this article as:
Bourguin, S. & Tudor, C.A. J Theor Probab (2012) 25: 536. doi:10.1007/s10959-010-0302-y

Abstract

We study the asymptotic behavior as n→∞ of the sequence
$$S_{n}=\sum_{i=0}^{n-1}K\bigl(n^{\alpha}B^{H_{1}}_{i}\bigr)\bigl(B^{H_{2}}_{i+1}-B^{H_{2}}_{i}\bigr)$$
where \(B^{H_{1}}\) and \(B^{H_{2}}\) are two independent fractional Brownian motions, K is a kernel function and the bandwidth parameter α satisfies certain hypotheses in terms of H1 and H2. Its limiting distribution is a mixed normal law involving the local time of the fractional Brownian motion \(B^{H_{1}}\). We use the techniques of the Malliavin calculus with respect to the fractional Brownian motion.

Keywords

Limit theorems Fractional Brownian motion Multiple stochastic integrals Malliavin calculus Regression model Weak convergence 

Mathematics Subject Classification (2000)

60F05 60H05 91G70 

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.SAMMUniversité de Paris 1 Panthéon-SorbonneParisFrance
  2. 2.Laboratoire Paul PainlevéUniversité de Lille 1Villeneuve d’AscqFrance